Various physical models are used to produce calculations of primary formulations and corrections for paints, e.g. according to Kubelka-Munk and Giovanelli, enabling the optical properties of the paints to be calculated and simulated. These optical properties are determined essentially by the nature of the pigments contained in the paint. A distinction can be drawn between isotropically and anisotropically light-scattering pigments. The isotropically light-scattering pigments cause multiple scattering of the incident light, so that the intensity of the reflected or scattered light is independent of both the direction of irradiation of the incident light and also the direction of observation. As a result of the multiple scattering the intensity of the reflected and scattered light is at least approximately isotropic. By contrast, the intensity of the reflected light in the case of anisotropically light-scattering pigments is dependent on the direction of irradiation of the incident light and on the direction of observation, as will be shown in more detail hereinafter.
A physical model used for calculating and simulating the optical properties of paints uses a so-called radiation transport equation (RTE) in which the light intensity depending on the directions of scattering of the light-scattering pigments of the paint is described in the form of a phase function. In the case of the isotropically light-scattering pigments as used in conventional paints, the phase function is a constant. Therefore, the radiation transport equation, which is dependent on the angle and layer thickness, can be described approximately by a linear differential equation system with constant coefficients and can be solved easily and efficiently in terms of computing time using an eigenvalue approach. The radiation transport equation is thus replaced by a linear differential equation system. However, the approximate simplification of these physical models can only be used for pigments with isotropically light-scattering properties.
In contrast to the conventional paints, so-called special effect paints additionally contain, besides the pigments with isotropically light-scattering properties, anisotropically light-scattering pigments which impart anisotropically light-scattering properties to the corresponding paints. These include aluminium and/or interference pigments such as for example mica particles or mica, which produce a so-called “pearlescent” effect in car paints, for example.
As the use of anisotropically light-scattering pigments results in a variable light intensity distribution of the reflected or scattered light when the corresponding coat of paint is irradiated, which is dependent not only on the direction of illumination and observation but also on an optical layer thickness, the phase function is not constant in this case, in contrast to the purely isotropically light-scattering pigments. For special effect paints an angle-dependent phase function thus has to be applied but then there is no longer an easily solvable linear differential equation system available. This can no longer be solved as a closed equation and therefore significantly greater numerical computing time has to be expended.